On some indices of foliations and applications

In this paper, we establish a relationship between the Milnor number, the χ-number and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont–Seade–Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: We provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau–Lins Neto and a new proof of a Soares’s inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.